In a series of papers, over the years, we have studied gravity and Yang-Mills theories in a formalism we have named the \(lc_2\)  formalism. In this framework, the (super)Poincaré algebra is spanned on just the physical degrees of freedom and one of the light-cone directions is identified as the evolution (“time”) variable. By doing this, the conjugate light-cone momentum is the Hamiltonian and can be read off by constructing the generators of the algebra order by order in the coupling constant [1]}. The formalism for gravity is set up on a Minkowski background and the price to pay is that the Hamiltonian is an infinite series of terms which after the four-point coupling becomes unmanageable. This means that we never fully prove the symmetries we realize but in our experience, checking symmetries up to the three-point coupling is sufficient to be convinced that the theory in question satisfies that particular symmetry. (We do know them all to the four-point coupling order.) Since this is a perturbative expansion in the gravity fields we can only trust it for weak enough fields.
